On the Almost Everywhere Stability of Discrete-Time Dynamical Systems

TL;DR

This paper extends the almost everywhere stability results of discrete-time dynamical systems from equilibria to invariant sets using Lyapunov densities and operator duality, removing some restrictive assumptions.

Contribution

It generalizes stability results to invariant sets and eliminates unnecessary assumptions like local stability and compactness in the discrete-time case.

Findings

Generalized stability from equilibria to invariant sets

Used duality between Frobenius-Perron and Koopman operators

Removed redundant assumptions in stability analysis

Abstract

For a dynamical system, it is known that the existence of a Lyapunov-type density function, called Lyapunov density or Rantzer's density function, implies convergence of Lebesgue almost all solutions to an equilibrium. Using the duality between Frobenius-Perron and Koopmann operators, we generalize this result from equilibrium to invariant sets, both for continuous- and discrete-time. Furthermore, some redundant assumptions that exist in the literature of almost everywhere stability for discrete-time, such as the local stability of the attractor and the compactness of the state space, has been removed.